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Dynamics in a ratio-dependent predator–prey model with predator harvesting. (English) Zbl 1122.34035
The authors consider the following ratio-dependent predator-prey model with nonzero constant predator harvesting
\[ \begin{aligned} \dot x(t)=& x(t)(1-x(t))-\frac{ax(t)y(t)}{y(t)+x(t)},\\ \dot y(t)=& y(t)\left(-d+\frac{bx(t)}{y(t)+x(t)}\right)-h, \end{aligned} \] where all parameters \(a,b,d\) and \(h\) are positive constants; \(h\) represents the rate of predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation), the subcritical and supercritical Hopf bifurcation. These results show far richer dynamics compared to the model without harvesting and different dynamics compared to the model with nonzero constant rate of prey harvesting developed by D. Xiao and L. S. Jennings [SIAM Appl. Math. 65, 737–753 (2005; Zbl 1094.34024)]. Biologically, it is shown that nonzero constant rate of predator harvesting can prevent mutual extinction as a possible outcome of the predator-prey interaction.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
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